Boolean Cutting is not Working
Booleans are a volume-based operation

Literally, boolean is about doing simple maths between two volumes. Like A + B or A - B.
A plane has zero volume. So you're asking Blender to do the maths A - 0... Which equals A.

If you want to remove the end of the nose, extrude your plane so it contains all the parts you want to cut off:



The modifier must be set on the target final object

Your goal is to have your elephant without its nose, so it's the elephant object that should wear the boolean modifier: